Consider utility functions of time 1 consumption of the generalized power utility form [u^{i}(x)=frac{1}{b-1}left(gamma_{i}+b x ight)^{frac{b-1}{b}}, quad

Consider utility functions of time 1 consumption of the generalized power utility form

\[u^{i}(x)=\frac{1}{b-1}\left(\gamma_{i}+b x\right)^{\frac{b-1}{b}}, \quad \text { with } b otin\{0,1\} \text {, for all } i=1, \ldots, I \text {, }\]

where agent \(i\) has discount factor \(\delta_{i}\), for \(i=1, \ldots, I\). By relying on condition (4.4), show that the Pareto optimal sharing rule is linear with respect to the aggregate endowment. In an analogous way, prove the same result in the case of exponential utility functions \(u^{i}(x)=-\gamma_{i} \exp \left(-x / \gamma_{i}\right)\), for all \(i=1, \ldots, I\).